Root locus — Evans' method, rules for sketching: Question bank

Reminder of the two rules everything rests on:

Angle condition $∠ G (s) H (s) = \pm (2 k + 1) \times 180°$ with $k = 0, 1, 2, \dots$ decides the shape (which points are on the locus). The factor $(2 k + 1)$ just enumerates the odd multiples $180°, 540°, \dots$ (all the same physical angle mod $360°$ ); different $k$ let different branches satisfy it.
Magnitude condition $K = 1/∣ G (s) H (s) ∣$ decides the gain label at each point.

Intuition Why the denominator is

$n - m$ , and why the centroid is the average Look at the figures below. Far from the origin, every pole and zero looks like it sits at roughly the same spot — you cannot resolve their individual positions from a great distance. So $G (s) H (s) = \frac{N ( s )}{D ( s )}$ collapses to $\frac{1}{( s - σ _{a} ) ^{n - m}}$ : the leftover $n - m$ poles (poles minus the $m$ that zeros "cancel" at infinity) behave like a single pole of multiplicity $n - m$ planted at their balance point $σ_{a}$ . The angle condition on that lone effective pole, $(n - m) \times ∠ (s - σ_{a}) = \pm (2 k + 1) \times 180°$ , immediately gives the asymptote-angle formula. The balance point $σ_{a}$ is the average because matching the two leading terms of the polynomial long division of $D / N$ places the effective pole exactly at (sum of poles − sum of zeros)/(n−m).

True or false — justify

The root locus shows how the OPEN-loop poles move as

K

increases.

False — the open-loop poles are fixed (roots of

D (s)

, independent of

K

). The locus tracks the closed-loop poles, which start at the open-loop poles when

K = 0

and move away as

K

grows.

Every branch of the locus must end on a finite open-loop zero.

False — only

m

branches (the number of finite zeros, roots of

N (s)

) end on zeros; the remaining

n - m

branches escape to infinity along the asymptotes. A system with no finite zeros sends all

n

branches off to infinity.

The locus is always symmetric about the real axis.

True — the characteristic polynomial has real coefficients, so any complex closed-loop pole is accompanied by its conjugate. The map is a mirror image top-to-bottom.

If all open-loop poles are in the left half plane, the closed-loop system is stable for every

K > 0

False — as

K

grows, branches can migrate right and cross the imaginary axis (e.g.

1/ [s (s + 1) (s + 2)]

goes unstable at

K = 6

). Left-half open-loop poles guarantee nothing about large-gain behaviour.

The centroid

σ_{a}

is the point where the locus actually crosses the imaginary axis.

False —

σ_{a}

is only where the asymptotes intersect the real axis. The true

j ω

-crossing comes from Routh–Hurwitz and is usually a different value.

Adding a zero (as a PD controller does) can bend the locus back into the left half plane.

True — a finite zero (a root of

N (s)

) attracts a branch, changing the asymptote count

n - m

and often pulling escaping branches toward the stable side. This is exactly how PD control adds damping.

The angle condition and the magnitude condition each independently determine the whole locus.

False — the angle condition alone determines the shape (which points lie on the locus). The magnitude condition only labels those points with their

K

value; it draws nothing new.

Increasing

K

always makes the response faster and better damped.

False — higher

K

typically moves poles toward lower damping ratio (branches curl toward

j ω

). It can speed the response but reduce damping, causing overshoot or instability.

Spot the error

"A real point is on the locus if there are poles to its right." — what's wrong?

Two errors: it must be poles and zeros to the right, and the count must be odd, not merely nonzero. An even count (e.g. two poles to the right) puts the point off the locus.

"There are

m

branches because there are

m

zeros." — fix it.

The number of branches equals

n

, the number of open-loop poles (the degree of

D (s) + K N (s)

s

n

when

n \geq m

). Of these,

m

terminate on zeros and

n - m

run to infinity.

"The breakaway point is always the midpoint between two poles." — why is this only sometimes right?

Symmetry forces the breakaway (the real point where branches leave the axis) to lie between two real poles, but its exact location comes from solving

d K / d s = 0

. Only for a symmetric pair like

{0, - 2}

does it land on the arithmetic midpoint; a nearby zero shifts it.

K = 1/∣ G (s) H (s) ∣

can be negative, so the locus includes negative gains." — correct it.

For the standard root locus

K \geq 0

, and

∣ G (s) H (s) ∣

is a magnitude (always

\geq 0

), so

K \geq 0

automatically. Negative

K

traces a complementary locus using the angle condition

∠ = 0°

, a different diagram.

"At the breakaway point the gain

K

is zero." — what's the real statement?

At breakaway two branches meet, meaning the polynomial has a repeated root, so

K (s) = - D (s) / N (s)

hits a local extremum (

d K / d s = 0

) — not necessarily zero. For

1/ [s (s + 2)]

the breakaway gain is

K = 1

"The asymptote angles depend on where the poles are located." — why false?

The angles

θ_{a} = \pm (2 k + 1) \times 180°/ (n - m)

depend only on the count

n - m

, not pole positions. Pole positions affect the centroid

σ_{a}

(where asymptotes sit), not their slopes.

Why questions

Why is the point on the locus governed by the angle condition and not the magnitude?

Because

L (s) = - 1

splits into "magnitude

= 1

" and "angle

= 180°

". For any point, you can always choose a

K

to fix the magnitude, so magnitude never rules a point out — only the fixed angle requirement does.

Why do complex-conjugate poles contribute zero net angle to a real test point?

A conjugate pair sits symmetrically above and below the real axis. Their angle contributions to any real point are equal and opposite, so they cancel — which is why only real singularities matter for the real-axis rule.

Why does the far-field locus look like a single pole of multiplicity

n - m

at the centroid?

For

∣ s ∣ \to \infty

the individual poles and zeros are unresolvable, so

G (s) H (s) \approx 1/ (s - σ_{a})^{n - m}

after matching the two leading terms of the division

D (s) / N (s)

(see figure s02). The whole cluster blurs into one effective pole at their weighted average

σ_{a}

, and applying the angle condition to it yields the asymptote angles.

Why does the angle of departure formula start with

180°

It comes from applying the angle condition

∠ G (s) H (s) = 180°

to a test point infinitesimally close to the complex pole (see figure s03). The

180°

is the target; the sums of angles from other poles/zeros are the corrections needed to hit it.

Why do we use Routh–Hurwitz for the

j ω

crossing instead of the angle condition?

The angle condition tells us the shape but not which

K

pushes a pole exactly onto

j ω

. Routh–Hurwitz directly yields the critical

K

that makes a row vanish, and its auxiliary equation gives the crossing frequency

ω

— the stability boundary.

Why is "shape first, gain second" the correct mental order?

The geometry (real-axis segments, asymptotes, breakaways) is decided entirely by pole/zero positions via the angle condition — independent of

K

. Only once the path is drawn do you use the magnitude condition to stamp

K

-values along it.

Edge cases

What does the locus look like for a single open-loop pole and no zeros,

G (s) H (s) = 1/ (s + a)

One branch starting at

s = - a

, and since

n - m = 1

the lone asymptote is at

180°

, so the pole slides straight left along the real axis toward

- \infty

. Stable for all

K > 0

Two real poles at the same location (a repeated pole), e.g.

1/ (s + a)^{2}

— what happens at

K = 0^{+}

Both branches start at the double pole

s = - a

, then immediately break perpendicular to the real axis (the breakaway is the pole itself). They rise/fall vertically since

n - m = 2

gives asymptotes at

90°, 270°

A pole sitting exactly on the imaginary axis at the origin,

G (s) H (s) = 1/ s

— is the origin on the locus?

The origin is the start of the branch at

K = 0

(it's the open-loop pole). As

K

increases the closed-loop pole moves left off the axis, so the system is stable for

K > 0

despite the marginal open-loop pole.

What if an open-loop pole and zero cancel exactly, e.g.

G (s) H (s) = \frac{( s + 2 )}{( s + 2 ) ( s + 5 )}

The common factor

(s + 2)

cancels, so it is neither a locus start nor end — no branch runs between them; effectively

n

and

m

each drop by one and the locus is that of

1/ (s + 5)

. Caution: exact cancellation never happens in real hardware, so a near-cancelled pole hides a slow, poorly-controllable mode (a hidden pole) that the root locus of the reduced system won't show.

What if

n = m

(equal poles and zeros)?

Then

n - m = 0

: no branches escape to infinity, all

n

branches terminate on finite zeros. There are no asymptotes to draw. This occurs with proper-but-not-strictly-proper loops.

A zero placed to the right of all poles on the real axis — is the segment beyond that zero on the locus?

To the right of the rightmost singularity the count of poles+zeros to the right is zero (even), so that segment is off the locus. The rule counts everything to the right, zeros included.

If two branches meet on the real axis and both leave it, what is true about

K

there?

They meet at a breakaway point where

K

reaches a local maximum along the real axis (

d K / d s = 0

); increasing

K

further forces the poles complex, so they leave the axis as a conjugate pair.

Can the root locus ever exist entirely off the real axis with no real-axis segment?

Only if the real-axis odd-count rule is never satisfied — e.g. a pure complex-conjugate pole pair with no real poles or zeros. Then branches curve in the plane without touching the real axis except by symmetry considerations.

Connections

Root locus — Evans' method, rules for sketching — the parent this bank tests
Characteristic equation & closed-loop poles
Routh–Hurwitz stability criterion — the $j ω$ -crossing tool
PID / PD controller design — zeros as reshaping tools
Bode plot & frequency response
Nyquist criterion
Damping ratio and settling time
GNC control loops